On a class of Model Hilbert Spaces

We provide a detailed description of the model Hilbert space $L^2(\bbR; d\Sigma; \cK)$, were $\cK$ represents a complex, separable Hilbert space, and $\Sigma$ denotes a bounded operator-valued measure. In particular, we show that several alternative approaches to such a construction in the literature are equivalent. These spaces are of fundamental importance in the context of perturbation theory of self-adjoint extensions of symmetric operators, and the spectral theory of ordinary differential operators with operator-valued coefficients.


Introduction
The principal purpose of this note is to recall and elaborate on the construction of the model Hilbert space L 2 (R; dΣ; K) and related Banach spaces L p (R; wdΣ; K), p ≥ 1. Here K represents a complex, separable Hilbert space, Σ denotes a bounded operator-valued measure, and w is an appropriate scalar nonnegative weight function. This model Hilbert space is known to play a fundamental role in various applications such as, perturbation theory of self-adjoint operators, the theory of self-adjoint extensions of symmetric operators, and the spectral theory of ordinary differential operators with operator-valued coefficients (cf. the end of Section 2).
In Section 2 we describe in detail the construction of L 2 (R; dΣ; K) following the approach used in [24]. We actually will present a slight generalization to the effect that we now explicitly permit that the bounded operator T = Σ(R) in K has a nontrivial null space. In the last part of Section 2 we will show that our construction is equivalent to alternative constructions employed by Berezanskii [7, Sect. VII.2.3] and another approach originally due to Gel'fand-Kostyuchenko [22] and Berezanskii [7,Ch. V].
It is a somewhat curious fact that in the alternative construction due to Berezanskii [7, Sect. VII.2.3] one has a choice in the order in which one takes a certain completion and a quotient with respect to a semi-inner product. In fact, different authors frequently chose one or the other of these two different routes without commenting on the equivalence of these two possibilities. Thus, we prove their equivalence in Appendix A.
We now briefly comment on the notation used in this paper: Throughout, H and K denote separable, complex Hilbert spaces, the inner product and norm in H are denoted by (·, ·) H (linear in the second argument) and · H , respectively. The identity operator in H is written as I H . We denote by B(H) the Banach space of linear bounded operators in H. The domain, range, kernel (null space) of a linear operator will be denoted by dom(·), ran(·), ker(·), respectively. The closure of a closable operator S is denoted by S. The Borel σ-algebra on R is denoted by B(R).

Direct Integrals and the Construction of the
Model Hilbert Space L 2 (R; dΣ; K) In this section we describe in detail the construction of the model Hilbert space L 2 (R; dΣ; K) (and related Banach spaces L p (R; wdΣ; K), p ≥ 1, w an appropriate scalar nonnegative weight function) following (and extending) a method first described in [24].
Throughout this section we make the following assumptions: Hypothesis 2.1. Let µ denote a σ-finite Borel measure on R, B(R) the Borel σ-algebra on R, and suppose that K and K λ , λ ∈ R, denote separable, complex Hilbert spaces such that the dimension function R ∋ λ → dim(K λ ) ∈ N ∪ {∞} is µ-measurable.
Assuming Hypothesis 2.1, let S({K λ } λ∈R ) be the vector space associated with the Cartesian product λ∈R K λ equipped with the obvious linear structure. Elements of S({K λ } λ∈R ) are maps , g(λ)) K λ ∈ C is µ-measurable for all g ∈ M.
Moreover, M is said to be generated by some subset F , F ⊂ M, if for every g ∈ M we can find a sequence of functions h n ∈ lin.
The definition of M was chosen with its maximality in mind and we refer to Lemma 2.4 and for more details in this respect. An explicit construction of an example of M will be given in Theorem 2.8. Remark 2.3. The following properties are proved in a standard manner: (i) If f ∈ M, g ∈ S({K λ } λ∈R ) and g = f µ-a.e. then g ∈ M.
(iii) If φ is a scalar-valued µ-measurable function and f ∈ M then φf ∈ M.
(In particular, any orthonormal basis {e n (λ)} n∈N in K λ will satisfy (α) and (β).) Then setting one has the following facts: Next, let w be a µ-measurable function, w > 0 µ-a.e., and consider the spacė with its obvious linear structure. OnL 2 (R; wdµ; M) one defines a semi-inner product (·, ·)L 2 (R;wdµ;M) (and hence a semi-norm · L2 (R;wdµ;M) ) by That (2.4) defines a semi-inner product immediately follows from the corresponding properties of (·, ·) K λ and the linearity of the integral. Next, one defines the equivalence relation ∼, for elements f, g ∈L 2 (R; wdµ; M) by f ∼ g if and only if f = g µ-a.e.
is well-defined (i.e., independent of the chosen representatives of the equivalence classes) and actually an inner product. Thus, L 2 (R; wdµ; M) is a normed space and by the usual abuse of notation we denote its elements in the following again by f, g, etc. Moreover, L 2 (R; wdµ; M) is also complete:  [24]. Separability of L 2 (R; wdµ; M) is proved in [11,Sect. 7.1] (see also [5,Subsect. 4.3.2]). Remark 2.6. Clearly, the analogous construction then defines the Banach spaces L p (R; wdµ; M), p ≥ 1.
Having reviewed the construction of L 2 (R; wdµ; M) =´⊕ R w(λ)dµ(λ) K λ in connection with a scalar measure wdµ, we now turn to the case of operator-valued measures and recall the following definition (we refer, for instance, to [5,  (ii) Σ(·) is strongly countably additive (i.e., with respect to the strong operator topology in H), that is, Moreover, Σ(·) is called an (operator-valued ) spectral measure (or an orthogonal operator-valued measure) if additionally the following condition (iii) holds: and hence Σ(B) 1/2 ξ K ≤ T 1/2 ξ K , ξ ∈ K, (2.11) shows that ker(T ) = ker T 1/2 ⊆ ker Σ(B) 1/2 = ker(Σ(B)), B ∈ B(R). (2.12) We will use the orthogonal decomposition and . Then T permits the 2 × 2 block operator representation with respect to the decomposition (2.13). By (2.12) one concludes that Σ(B), B ∈ B(R), is necessarily of the form , for some 0 ≤ Σ 1 (B) ∈ B(K 1 ), D ∈ B(K 0 , K 1 ), (2.15) with respect to the decomposition (2.13). The computation yields D = 0 as f 0 ∈ K 0 was arbitrary. Thus, Σ(B), B ∈ B(R), is actually also of diagonal form , for some 0 ≤ Σ 1 (B) ∈ B(K 1 ), (2.17) with respect to the decomposition (2.13). Moreover, let µ be a control measure for Σ (equivalently, for Σ 1 ), that is, The following theorem was first stated in [24] under the implicit assumption that Σ(R) = T = I K . In this paper we now treat the general case T ∈ B(K), in particular, we explicitly permit the existence of a nontrivial kernel of T : and ker(Λ) = ker(T ), (2.20) so that the following assertions (i)-(iii) hold: in particular, where {e n } n∈I denotes any sequence of linearly independent elements in K with the property lin.span{e n } n∈I = K. In particular, where (cf. (2.14) and (2.17))
Proof. Since the current version of this theorem extends the earlier one in [24], we now repeat it for the convenience of the reader. Moreover, we will shed additional light on the proof of (2.23), correcting an oversight in this connection in [24].
By considering only rational linear combinations (i.e., Hence we can define a semi-inner product (·, ·) λ on V, for all v = n α n e n , w = n β n e n ∈ V.
Next, let K λ be the completion of V/N λ with respect to · λ , where and define (for convenience) Again we identify an element v ∈ V with an element in V/N λ ⊆ K λ . Applying Lemma 2.4, the collection {e n } n∈I then generates a measurable family of Hilbert Hence we can defineΛ and denote by Λ ∈ B K, L 2 (R; dµ; M Σ ) , with the closure ofΛ. In particular, one obtains and hence ker(Λ) = ker(T ) = K 0 .
(2.36) Then properties (i) and (ii) hold and we proceed to illustrating property (iii): Introduce the operator (Alternatively, this follows from the fact that ker T Hence, also S(B) has a bounded extension (its closure) to all of K with Moreover, one computes In particular, this also yields 44) and note that S(R) = I K .
(2.45) Then one obtains [56, Theorem 4.19 (b)]). By the fact that S(B) ∈ B(K) and hence S(B) ∈ B(K), one concludes that T Thus, taking adjoints in (2.43) one obtains (2.48) Let ξ ∈ K, then combinng (2.21), (2.22), and (2.48) yields Implicitly in the proof of Theorem 2.8 is a special case of the following result, which appears to be of independent interest. It may well be known, but since we could not quickly find it in the literature we include its short proof for the convenience of the reader: Lemma 2.9. Let H be a complex, separable Hilbert space, F, G self-adjoint operators in H, and 0 ≤ F ≤ G. Then (2.50) In particular, if in addition F, G ∈ B(H) and ker(G) = {0}, then and hence obtain the 2 × 2 block operator representations with respect to the decomposition (2.54), with self-adjoint operators F 1 , G 1 in H 1 satisfying 0 ≤ F 1 ≤ G 1 . In particular, Then for any y ∈ dom(F α 1 ) and x ∈ ran(G α 1 ) = dom(G −α 1 ), one computes using self-adjointness of F α 1 and (2.57) and hence (2.50) since also ran(F α 1 ) = ran(F α ). The fact (2.51) then follows from the closed graph theorem.
Next, we recall that the construction in Theorem 2.8 is essentially unique: (2.59) Remark 2.11. (i) Without going into further details, we note that M Σ depends of course on the control measure µ. However, a change in µ merely effects a change in density and so M Σ can essentially be viewed as µ-independent.
(ii) With 0 < w a µ-measurable weight function, one can also consider the Hilbert space L 2 (R; wdµ; M Σ ). In view of our comment in item (i) concerning the mild dependence on the control measure µ of M Σ , one typically puts more emphasis on the operator-valued measure Σ and hence uses the more suggestive notation L 2 (R; wdΣ; K) instead of the more precise L 2 (R; wdµ; M Σ ) in this case.
Next, let V = lin.span{e n ∈ K | n ∈ I}, V = K, (2.60) and define (2.61) The fact that {e n } n∈I generates M Σ then implies that V Σ is dense in the Hilbert space L 2 (R; dµ; M Σ ), that is, Since the operator-valued distribution function Σ(·) has at most countably many discontinuities on R, denoting by S Σ the corresponding set of discontinuities of Σ(·), introducing the set of intervals the minimal σ-algebra generated by B Σ coincides with the Borel algebra B(R).
Hence one can introduce V Σ = lin.span χ (α,β] e n ∈ L 2 (R; dµ; M Σ ) α, β ∈ R\S Σ , n ∈ I , (2.64) which still retains the density property in (2.62), that is, In the following we briefly describe an alternative construction of L 2 (R; dΣ; K) used by Berezanskii [7, Sect. VII.2.3] in order to identify the two constructions. Introduce On C 0,0 (R; K) one can introduce the semi-inner product where the integral on the right-hand side of (2.67) is well-defined in the Riemann-Stieltjes sense. Introducing the kernel of this semi-inner product by In particular, and (cf. also [40, Corollary 2.6]) (2.70) extends to piecewise continuous K-valued functions with compact support as long as the discontinuities of u and v are disjoint from the set S Σ (the set of discontinuities of Σ(·)). Since Kats' work in the case of a finite-dimensional Hilbert space K (cf. [35], [36] and also Fuhrman [21, Sect. II.6] and Rosenberg [51]), and especially in the work of Malamud and Malamud [40], who studied the general case dim(K) ≤ ∞, it has become customary to interchange the order of taking the quotient with respect to the semi-inner product and completion in this process of constructing L 2 (R; dΣ; K). More precisely, in this context one first completes C 0,0 (R, K) with respect to the semi-inner product (2.67) to obtain a semi-Hilbert space and then takes the quotient with respect to the kernel of the underlying semi-inner product, as described in method (I) of Appendix A. Berezanskii's approach in [7, Sect. VII.2.3] corresponds to method (II) discussed in Appendix A. The equivalence of these two methods is not stated in these sources and hence we spelled this out explicitly in Lemma A.1 in Appendix A. Next we will indicate that Berezanskii's construction of L 2 (R; dΣ; K) (and hence the corresponding construction by Kats (if dim(K) < ∞) and by Malamud and Malamud (if dim(K) ≤ ∞) is equivalent to the one in [24] and hence to that outlined in Theorem 2.8: Theorem 2.12. The spaces L 2 (R; dΣ; K) and L 2 (R; dµ; M Σ ) are isometrically isomorphic.
Proof. We first recall that the set V Σ in (2.64) is dense in L 2 (R; dµ; M Σ ). On the other hand, it was shown in the proof of Theorem 2.14 in [40] that V Σ is also dense in L 2 (R; dΣ; K). The fact then establishes a densely defined isometry between the Hilbert spaces L 2 (R; dΣ; K) and L 2 (R; dµ; M Σ ) which extends by continuity to a unitary map.
As a result, dropping the additional "hat" on the left-hand side of (2.69), and hence just using the notation L 2 (R; dΣ; K) for both Hilbert space constructions is consistent.
We continue this section by yet another approach originally due to Gel'fand and Kostyuchenko [22] [39], [40]: Introducing an operator K ∈ B 2 (H) with ker(K) = ker(K * ) = {0}, one has the existence of the weakly µ-measurable nonnegative operator-valued function Ψ K (·) with values in B 1 (H), such that In fact, the derivative Ψ K (·) exists in the B 1 (H)-norm (cf. [10] and [39], [40]). Introducing the semi-Hilbert space H t , t ∈ R, as the completion of dom K −1 with respect to the semi-inner product factoring H t by the kernel of the corresponding semi-norm ker( · Ht ) then yields the Hilbert space H t = H t / ker( · Ht ), t ∈ R. One can show (cf. [39], [40]) that yielding yet another construction of L 2 (R; dΣ; K). We conclude this section by sketching some applications to the perturbation theory of self-adjoint operators and to the theory of self-adjoint extensions of symmetric operators, following [24]. We will also briefly comment on work in preparation concerning the spectral theory of ordinary differential operators with operator-valued coefficients.
We start by recalling the following result: The family of strongly right-continuous orthogonal spectral projections { E L (λ)} λ∈R of H L in H L is given by For a variety of additional results in this context we refer to [24].
(II) Self-adjoint extensions of symmetric operators.
and introducing the weight function and Hilbert space L 2 (R; Thus, the linear maṗ where Introducing the control measureμ(B) = n∈I 2 −n (u n , Ω(B)u n ) N , B ∈ B(R), and Λ as in Theorem 2.8, we may define L p (R; wd Ω; N ), p ≥ 1, w ≥ 0 an appropriate weight function. Of special importance in this section are weight functions of the type w r (λ) = (1 + λ 2 ) r , r ∈ R, λ ∈ R. In particular, introducing
Assuming (a, b) ⊆ R, and supposing that V : (a, b) → B(H) is a weakly measurable operator-valued function, we recently developed Weyl-Titchmarsh theory for certain self-adjoint operators H α in L 2 ((a, b); dx; H) associated with the operator-valued differential expression τ = −(d 2 /dx 2 ) + V (·) (cf. [30]). These are suitable restrictions of the maximal operator H max in L 2 ((a, b); dx; H) defined by In particular, assuming in addition that a is a regular endpoint for τ and b is of limit-point type for τ , the operator H α defined as the restriction of H max by dom(H α ) = {u ∈ dom(H max ) | sin(α)u ′ (a) + cos(α)u(a) = 0}. (2.131) where α = α * ∈ B(H) is self-adjoint in L 2 ((a, b); dx; H). Conversely, all self-adjoint restrictions of H max arise in this manner. Introducing θ α (z, ·, x 0 , ), φ α (z, ·, x 0 ) as those B(H)-valued solutions of τ Y = zY which satisfy the initial conditions 132) one of the principal results in [30] establishes the existence of B(H)-valued Weyl-Titchmarsh solutions ψ α (z, ·) of τ Y = zY of the form valid in the strong sense in H. The function m α (·) contains all the spectral information of H α and is closely related to the Green's function of H α as discussed in [30]. Introducing the Hilbert space L 2 (R; dΩ α ; H), the spectral representation (resp., model representation) of H α then aims at exhibiting the unitary equivalence of H α with the operator of multiplication by the independent variable in L 2 (R; dΩ α ; H). Under stronger hypotheses on V than those recorded in (2.129), for instance, continuity of V (·) in B(H), such a result has been shown by Rofe-Beketov [50] and Gorbačuk [32], and subsequently, under hypotheses close to those in (2.129), by Saito [52]. Our own approach to this circle of ideas is in preparation [31].
We conclude this section by noting that certain classes of unbounded operatorvalued potentials V lead to applications to multi-dimensional Schrödinger operators in L 2 (R n ; d n x), n ∈ N, n ≥ 2. It is precisely the connection between multidimensional Schrödinger operators and one-dimensional Schrödinger operators with unbounded operator-valued potentials which originally motivated our interest in this area.

Appendix A. Completion of Semi-Metric Spaces
In this appendix we establish the equivalence of two approaches to the procedure of completion of semi-metric spaces. For background material on semi-metric spaces we refer to [57,Ch. 9].
We start by recalling that a semi-metric space (S, ρ) is a set S with a semi-metric ρ(x, y) ≤ ρ(x, z) + ρ(z, y), x, y, z ∈ S.
We point out that a semi-metric space may have two distinct elements x, y ∈ S with ρ(x, y) = 0. Nevertheless, one can introduce the notion of Cauchy sequences and limits in a semi-metric space (S, ρ) as usual. Of course, in this case convergent sequences may have several distinct limits. A semi-metric space (S, ρ) is called complete if every Cauchy sequence of points in S has a limit which also lies in S. Next, we discuss two approaches to completion of a semi-metric space.
(I) Given a semi-metric space (S, ρ), one introduces a semi-metric space S 1 , ρ 1 , where S 1 is the set of all Cauchy sequence in S and It follows from the triangle inequality (A.3) for ρ that |ρ(x(n), y(n)) − ρ(x(m), y(m))| ≤ ρ(x(n), x(m)) + ρ(y(n), y(m)), (A. 5) and hence for all x, y ∈ S 1 the sequence {ρ(x(n), y(n))} n∈N is Cauchy in R. Thus, the limit in (A.4) exists, and using (A.4) one verifies that ρ 1 is a semi-metric on S 1 . Moreover, it has been shown in [57, p. 176] that S 1 , ρ 1 is a complete semi-metric space and that (S, ρ) is isometric to a dense subset in S 1 , ρ 1 . Introducing an equivalence relation ∼ on S 1 by x ∼ y whenever ρ 1 ( x, y) = 0, x, y ∈ S 1 , (A.6) one defines the set of equivalence classes and a metric on S 1 by It follows from the triangle inequality for ρ 1 that ρ 1 ( x 1 , y 1 ) = ρ 1 ( x 2 , y 2 ) whenever x 1 ∼ x 2 and y 1 ∼ y 2 , (A.9) and hence ρ 1 is a well-defined metric on S 1 . Thus, (S 1 , ρ 1 ) is a complete metric space, and (S, ρ) is isometric to a dense subset of (S 1 , ρ 1 ). Now, we consider a different approach: (II) Given a semi-metric space (S, ρ), define an equivalence relation ∼ on S by It follows from the triangle inequality for ρ that for all x 1 , x 2 , y 1 , y 2 ∈ S ρ(x 1 , y 1 ) = ρ(x 2 , y 2 ) whenever x 1 ∼ x 2 and y 1 ∼ y 2 . (A.14) Thus, S 2 , ρ 2 is a complete semi-metric space. Introducing an equivalence relation ∼ on S 2 by x ∼ y whenever ρ 2 ( x, y) = 0, x, y ∈ S 2 , (A.15) one defines the set of equivalence classes and a metric on S 2 by Again, it follows from the triangle inequality for ρ 2 that ρ 2 is a well-defined metric on S 2 . Thus, (S 2 , ρ 2 ) is a complete metric space, and (M, d) and hence (S, ρ) are isometric to a dense subset of (S 2 , ρ 2 ). The main result of this appendix is the following isometry lemma.
Proof. To establish an isometric bijection T : S 1 → S 2 one proceeds as follows: For every element x = {x(n)} n∈N ∈ S 1 one defines x ∈ S 2 by x = {[x(n)] ρ } n∈N . Then  (i.e., J is well-defined). It follows from the above constructions that the domain of J is all of S 1 and the range is all of S 2 . Moreover, since ρ 1 is a metric, (A.18) implies that J is one-to-one and hence a bijection.